Related papers: Generating mapping class groups with elements of f…

For a finite group $G$, the vertices of the prime graph $\Gamma(G)$ are the primes that divide $|G|$, and two vertices $p$ and $q$ are connected by an edge if and only if there is an element of order $pq$ in $G$. Prime graphs of solvable…

The mapping class group $\mathrm{Mod}_{g, 1}$ of a surface with one marked point can be identified with an index two subgroup of $\mathrm{Aut}(\pi_1 \Sigma_g)$. For a surface of genus $g \geq 2$, we show that any action of $\mathrm{Mod}_{g,…

Let ${\rm GK}(G)$ be the prime graph associated with a finite group $G$ and $D(G)$ be the degree pattern of $G$. A finite group $G$ is said to be $k$-fold OD-characterizable if there exist exactly $k$ non-isomorphic groups $H$ such that…

We study finite p-groups G of coclass upto 4 for which the group Aut_z(G) of all central automorphisms of G is of minimal possible order. As a consequence, we obtain very short and elementary proofs of main results of Sharma and Gumber [7].

We prove that the extended mapping class group is generated by three orientation reversing involutions.

Let f be a Z/2Z-spin structureon a closed surface S of genus g>3. We determine a generating set of the stabilizer of f in the mapping class group of S consisting of Dehn twists about an explicit collection of 2g+1 curves in S. If g=3 then…

We prove that the invariably generating graph of a finite group can have an arbitrarily large number of connected components with at least two vertices.

We obtain the formula computing the number of isomorphic classes of element systems with characters over finite commutative group $G$.

Let $\mathcal{X}$ be a smooth Fano threefold over the complex numbers of Picard rank $1$ with finite automorphism group. We give numerical restrictions on the order of the automorphism group $\mathrm{Aut}(\mathcal{X})$ provided the genus…

We show that the natural map from the mapping class groups of surfaces to the automorphism groups of free groups, induces an infinite loop map on the classifying spaces of the stable groups after plus construction. The proof uses…

For any positive integer $n$, we exhibit a cofinite subgroup $\Gamma_n$ of the mapping class group of a surface of genus at most two such that $\Gamma_n$ admits an epimorphism onto a free group of rank $n$. We conclude that…

Let G be a cyclic p-group of order p^n acting by automorphisms on a (non-necessarily commutative) ring R. Suppose there is an element x in R such that (1 + t + ... + t^{p-1})(x) = 1, where t is an element of order p in G. We show how to…

Let $G$ be a finite group. The co-prime order graph of $G$ is the graph whose vertex set is $G$, and two distinct vertices $x,y$ are adjacent if gcd$(o(x),o(y))$ is either $1$ or a prime, where $o(x)$ and $o(y)$ are the orders of $x$ and…

The genus spectrum of a finite group $G$ is a set of integers $g \geq 2$ such that $G$ acts on a closed orientable compact surface $\Sigma_g$ of genus $g$ preserving the orientation. In this paper we complete the study of spectrum sets of…

In \cite{HigherGnk}, the author has constructed natural maps from fundamental groups of topological spaces (restricted configuration spaces) to the groups $G_{n}^{k}$. In the present paper, we show that in the case of $n=k+1$, the group…

We construct a minimal generating set of the level 2 mapping class group of a nonorientable surface of genus $g$, and determine its abelianization for $g\ge4$.

We determine all posible orders of automorphisms of finite order of complex K3 surfaces or of K3 surfaces in characteristic $p>3$. E.g., a positive integer $N$ is the order of an automorphism of a complex K3 surface if and only if…

In this work we compute the first integral cohomology of the pure mapping class group of a non-orientable surface of infinite topological type and genus at least 3. To this purpose, we also prove several other results already known for…

Let $X$ be a compact Riemann surface of genus $g\geq 2$. Let $Aut(X)$ be its group of automorphisms and $G\subseteq Aut(X)$ a subgroup. Sharp upper bounds for $|G|$ in terms of $g$ are known if $G$ belongs to certain classes of groups, e.g.…

Let $S$ be a closed oriented surface and $G$ a finite group of orientation preserving automorphisms of $S$ whose orbit space has genus at least $2$. There is a natural group homomorphism from the $G$-centralizer in $Diff^+(S)$ to the…